## Table of Contents

## Prefixes

Prefixes are useful for expressing units of physical quantities that are either very big or very small. If you write out such big or small numbers, it will be time consuming and prone to errors.

- In the English language, prefixes are letters which are placed in front of words to make new words.

Some of the Greek prefixes and their symbols to indicate decimal sub-multiples and multiples of the SI units are:

Number | Prefix | Number | Prefix |
---|---|---|---|

$10^{-9}$ | nano (n) | $10^{-1}$ | deci (d) |

$10^{-6}$ | micro ($\mu$) | $10^{3}$ | Kilo (K) |

$10^{-3}$ | milli (m) | $10^{6}$ | Mega (M) |

$10^{-2}$ | centi (c) | $10^{9}$ | Giga (G) |

## Common Examples Of Prefixes

- 1 cm = 0.01 metre
- 1 kg = 1000 grams
- 1 GB = 1 000 000 000 bytes
- 0.000 0052 s = $5.2 \, \mu \text{s}$ = $5.2 \times 10^{-6} \, \text{s}$

## Standard Form & Order Of Magnitude

$5.2 \times 10^{-6} \, \text{s}$ is what is known as **standard form**. When numbers are too large or too small, it is convenient to express them in standard form in the following format:

$$\text{M} \times 10^{\text{N}}$$

, where:

$\text{M}$ is in the range of: $1 \leq \text{M} < 10$

$\text{N}$ denotes the **order of magnitude** and is an integer.

Order of magnitude are used to estimate numbers which are extremely large to the nearest power of ten. The table below shows how orders of magnitude are used to compare base quantities – mass and length.

Mass ($\text{kg}$) | Factor | Length ($\text{m}$) | Factor |
---|---|---|---|

Electron | $10^{-30}$ | Radius of proton | $10^{-15}$ |

Proton | $10^{-27}$ | Radius of atom | $10^{-10}$ |

Ant | $10^{-3}$ | Height of ant | $10^{-3}$ |

Human | $10^{1}$ | Height of human | $10^{0} = 1$ |

Earth | $10^{24}$ | Radius of Earth | $10^{7}$ |

Sun | $10^{30}$ | Radius of Sun | $10^{9}$ |

Order of magnitude is useful when you do “back-of-the-envelope” calculations. A back-of-the-envelope calculation is a rough calculation, typically jotted down on any available scrap of paper such as an envelope.

Back-of-the-envelope calculations are used to quickly check something. If your back-of-the-envelope calculations yield several orders of magnitude bigger or smaller than what you expect, your formula or input variables must be wrong.

## Worked Examples

### Example 1

The length of the world’s smallest playable guitar is 13 micrometers ($13 \, \mu \text{m}$). Represent the guitar’s length in standard form.

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$$\begin{aligned} 13 \, \mu \text{m} &= 0.000013 \text{ m} \\ &= 13 \times 10^{-6} \text{ m} \\ &= 1.3 \times 10^{-5} \text{ m} \end{aligned}$$

The guitar’s length is $1.3 \times 10^{-5} \text{ m}$ in standard form.

### Example 2

What level of precision is implied in an order-of-magnitude calculation?

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Zero significant digits. An order-of-magnitude calculation is accurate only within a factor of 10.

### Example 3: Understanding Prefixes

What prefix would you use to express the weight of a bacterium that typically weighs about 0.000 000 003 grams, and what would that weight be in the prefixed unit?

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The prefix for $10^{-9}$ is nano (n), so the weight of the bacterium would be expressed as 3 nanograms (3 ng).

### Example 4: Converting Units

Convert 0.045 kilometers to meters using the appropriate prefix conversion.

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The prefix for $10^3$ is Kilo (K), indicating a factor of 1000. Therefore, 0.045 kilometers is equal to $0.045 \times 1000 = 45 \text{ meters}$.

### Example 5: Deciphering Common Examples

Given that 1 GB = 1,000,000,000 bytes, express this in terms of the appropriate Greek prefix for the number of bytes.

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The prefix for $10^9$ is Giga (G), so 1 GB is correctly expressed as 1 Gigabyte, indicating 1,000,000,000 bytes.

### Example 6: Standard Form Calculation

Express the mass of the Earth, which is $5.972 \times 10^{24}$ kg, in standard form using the appropriate prefix for its magnitude.

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The appropriate prefix for $10^{24}$ is Yotta (Y), so the mass of the Earth could be referred to as 5.972 Yottakilograms (Ykg), although this is not a commonly used expression in practice.

### Example 7: Order of Magnitude Comparison

Compare the order of magnitude for the length of an ant ($10^{-3} \text{m}$) to the height of a human ($10^0 = 1 \text{ m}$) and explain what this tells us about their sizes in relation to each other.

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The order of magnitude for the length of an ant is $10^{-3}$, and for the height of a human is $10^0 = 1$. This indicates that the ant’s length is three orders of magnitude smaller than the human’s height, or in simpler terms, the ant is 1,000 times shorter than the human, illustrating a practical application of comparing sizes using orders of magnitude.